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A192041 Decimal approximation of x such that f(x)=1/2, where f is the Fibonacci function described in Comments.

%I A192041 #7 Feb 13 2014 13:23:50
%S A192041 4,5,0,7,0,6,6,6,6,5,7,4,5,4,4,6,0,0,2,3,0,6,0,5,0,6,3,1,4,0,3,2,8,5,
%T A192041 7,1,5,1,8,1,4,4,0,2,4,0,2,0,3,6,2,2,4,6,1,8,7,8,4,7,5,3,5,5,7,7,8,1,
%U A192041 6,3,5,8,9,8,9,0,4,0,4,7,9,9,3,5,5,7,5,9,8,7,3,2,9,4,1,0,4,3,4,3
%N A192041 Decimal approximation of x such that f(x)=1/2, where f is the Fibonacci function described in Comments.
%C A192041 f(x)=(r^x-r^(-x*cos[pi*x]))/sqrt(5), where r=(golden ratio)=(1+sqrt(5))/2.  This function, a variant of the Binet formula, gives Fibonacci numbers for integer values of x; e.g., f(3)=2, f(4)=3, f(5)=5.
%e A192041 0.450706666574544600230605063140328571518144024020
%t A192041 r = GoldenRatio; s = 1/Sqrt[5];
%t A192041 f[x_] := s (r^x - r^-x Cos[Pi x]);
%t A192041 x /. FindRoot[Fibonacci[x] == 1/2, {x, 5}, WorkingPrecision -> 100]
%t A192041 RealDigits[%, 10]
%t A192041 (Show[Plot[#1, #2], ListPlot[Table[{x, #1}, #2]]] &)[
%t A192041 Fibonacci[x], {x, -7, 7}]
%t A192041 (* _Peter J. C. Moses_, Jun 21 2011 *)
%Y A192041 Cf. A192038.
%K A192041 nonn,cons
%O A192041 0,1
%A A192041 _Clark Kimberling_, Jun 21 2011